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G.f. satisfies: A(x) = Product_{n>=1} (1+x^n)*(1 + x^n*A(x))/((1-x^n)*(1 - x^n*A(x))).
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%I #7 Mar 30 2012 18:37:27

%S 1,4,20,112,676,4328,28912,199392,1409364,10157828,74375640,551715264,

%T 4137527408,31318286632,238958947328,1835960454272,14192132860868,

%U 110298595778872,861338925309604,6755283201399776,53185599585579640

%N G.f. satisfies: A(x) = Product_{n>=1} (1+x^n)*(1 + x^n*A(x))/((1-x^n)*(1 - x^n*A(x))).

%C Related q-series (Heine) identity:

%C 1 + Sum_{n>=1} x^n*Product_{k=0..n-1} (y+q^k)*(z+q^k)/((1-x*q^k)*(1-q^(k+1)) = Product_{n>=0} (1+x*y*q^n)*(1+x*z*q^n)/((1-x*q^n)*(1-x*y*z*q^n)), here q=x, x=x, y=A(x), z=1.

%F G.f. satisfies: A(x) = 1 + Sum_{n>=1} x^n*Product_{k=0..n-1} (A(x) + x^k)*(1+x^k)/(1-x^(k+1))^2 due to the Heine identity.

%e G.f.: A(x) = 1 + 4*x + 20*x^2 + 112*x^3 + 676*x^4 + 4328*x^5 +...

%e The g.f. A = A(x) satisfies:

%e A = (1+x)*(1+x*A)/((1-x)*(1-x*A)) * (1+x^2)*(1+x^2*A)/((1-x^2)*(1-x^2*A)) * (1+x^3)*(1+x^3*A)/((1-x^3)*(1-x^3*A)) *...

%e A = {1 + 2*x*(A+1)/(1-x)^2 + 2*x^2*(A+1)*(A+x)*(1+x)/((1-x)*(1-x^2))^2 + 2*x^3*(A+1)*(A+x)*(A+x^2)*(1+x)*(1+x^2)/((1-x)*(1-x^2)*(1-x^3))^2 +...

%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=prod(k=1,n,(1+x^k*A)*(1+x^k)/((1-x^k+x*O(x^n))*(1-x^k*A))));polcoeff(A,n)}

%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,n,x^m*prod(k=0,m-1,(A+x^k)*(1+x^k)/(1-x^(k+1)+x*O(x^n))^2)));polcoeff(A,n)}

%Y Cf. A192625, A192620, A192622.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jul 06 2011